Several marketplaces called “bitcoin exchanges” allow people to buy or sell bitcoins using different currencies. Mt. Gox is the largest bitcoin exchange.
Exchanges are really just a medium between traders. People can set limit orders to buy / sell bitcoins for a certain price. The exchange will match buyers and sellers when conditions of both the buyer and the seller are met.
Other way to describe an exchange would be like this -An exchange is where buyers and sellers conduct their business. A seller of BTC deposits BTC with the exchange’s address. He can then use his positive BTC balance in the exchange to sell his BTC for Dollars (or other coins). Similarly, a buyer of BTC deposits USD with the exchange and then uses the balance to buy BTC from sellers.